Maximal non-nonnil-principal ideal rings

Abstract

Let [Formula: see text] be the set of all commutative rings with unity whose nilradical is a divided prime ideal. The concept of maximal non-nonnil-PIR is introduced to generalize the concept of maximal non-PID. A ring extension [Formula: see text] in [Formula: see text] is a called a maximal non-nonnil-principal ideal ring if [Formula: see text] is not a nonnil-principal ideal ring but each subring of [Formula: see text] properly containing [Formula: see text] is a nonnil-principal ideal ring. It is shown that [Formula: see text] (respectively, [Formula: see text] is a maximal non-nonnil-PIR subring of [Formula: see text] (respectively, [Formula: see text]) if and only if [Formula: see text] (respectively, [Formula: see text] is a maximal non-PID subring of [Formula: see text] (respectively, [Formula: see text]).